DTIC ADA413449: A High Efficiency Algorithm for Combinatorial Synthesis: Maximum Utilization of Minimal Libraries

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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
A High Efficiency Algorithm for Combinatorial Synthesis: Maximum Utilization of F49620-00-1-0297
Minimal Libraries
6. AUTHOR(S)
Herschel Rabitz
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
REPORT NUMBER
Princeton University AF 6773-3
Department of Chemistry
Princeton, NJ 08544
9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ESI 10. SPONSORING / MONITORING
AGENCY REPORT NUMBER
Air Force Office of Scientific Research
AFOSR/NM
801 North Randolph Street, Room 732
Arlington, VA 22203-1977
11. SUPPLEMENTARY NOTES
12a. DISTRIBUTION IAVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 Words)
This research addressed the common mathematical characteristics of a broad variety of science and engineering applications having
large numbers of input variables for exploration. A family of High Dimensional Model Representation (HDMR) techniques was
developed to provide efficient and accurate input-output maps for such problems. HDMR operates by breaking down the original high
dimensional map into a hierarchy of interrelated lower dimensional components. The mathematical and numerical foundations of
HDMR were developed, along with demonstrations of its capabilities in selected chemistry, physics, and engineering applications.
20030513 0*69
14. SUBJECT TERMS 15. NUMBER OF PAGES
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High dimensional model representation, Functional mapping, Interpolation, Molecular properties 16. PRICE CODE
17. SECURITY CLASSIFICATION 1 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
LI
NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Sid. Z39-18
298-102
A HIGH EFFICIENCY ALGORITHM FOR COMBINATORIAL SYNTHESIS:
MAXIMUM UTILIZATION OF MINIMAL LIBRARIES
F49620-00-1-0297
Herschel Rabitz
Department of Chemistry
Princeton University
Objectives
Many problems in the science and engineering disciplines have very large numbers of
input variables. A common necessity is an exploration of these high dimensional spaces,
with the effort conventionally thought to grow exponentially with the number of variables
in a prohibitive manner. This research developed a special set of high dimensional model
representation (HDMR) tools specifically honed for treating such problems. The HDMR
technique is capable of reducing the sampling effort from being exponential down to only
polynomial scaling in the number of variables. Applications of the concept abound both
in mathematical modeling, as well as in the direct treatment of high dimensional field-
generated and industrial data sets. The research developed the mathematical and
numerical concepts of HDMR, as well as demonstrated its capabilities in selected
applications. These applications originally were focused on enhancing the capability of
discovering new molecular materials, but they have also been extended to other
engineering and scientific domains.
Summary of the Research Effort During the Period of the Grant
During the period of this grant, many of the basic conceptual and mathematical
foundations of HDMR were set forth. The underlying concept was to break down a high
dimensional input-output problem into a set of interconnected low dimensional
subspaces. By using suitable metrics and definitions of orthogonality, representations for
the individual low dimensional components may be determined from suitable modeling
or laboratory/field data, ultimately resulting in a full HDMR which captures the input-
output behavior of the entire high dimensional system. Techniques were developed for
sampling the input variable space, ranging from utilizing ordered to randomly located
data points. The scaling of these various procedures in terms of effort was explored, with
mathematical arguments and evidence indicating that dramatic savings (even up to orders
of magnitude in effort) can be achieved with these techniques.
Accomplishments
A. Theoretical and Algorithmic Foundations of High Dimensional Model
Representations (HDMR).
(1) High Dimensional Model Representations[l]. An important concern when
utilizing HDMR is the number of experiments or modeling excursions necessary to
effectively learn the system input -+ output behavior, which is typically a non-
linear relationship. Although simple logic suggests that the number of runs could
grow exponentially with the number of input variables, broad evidence indicates
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that the required effort often scales far more comfortably. This work reviewed the
emerging family of HDMR concepts and techniques capable of dealing with such
input -- output problems in a practical fashion. A summary of the state of the
subject was presented.
(2) Efficient Implementation of High Dimensional Model Representations[2]. The
main assumption of HDMR is that for most well defined physical systems the
output may be approximated by a sum of hierarchical functions whose
dimensionality is much smaller than the dimensionality of the full system. The
formulation of HDMR in this work assumed that the data was randomly scattered
throughout the domain of the input. It was shown that the number of samples
needed for output representation to a given tolerance was invariant to the
dimensionality of the input variable space, thereby providing for a very efficient
means to perform high dimensional interpolation. Selected applications of HDMR!
s were presented from sensitivity analysis and time-series analysis.
(3) High Dimensional model Representations Generatedf rom Low Dimensional Data
Samples I: MP-Cut HDMR[3]. For a high dimensional system, an output f(x) is
commonly a function of many input variables x={xl,x2, .,Xn } with n -102or
larger. When the approximations given by the first and the second order HDMR
correlated functions are not adequate, this work introduced a monomial-based
preconditioned HDMR method to represent the higher order terms of a HDMR
expansion. The accuracy of the HDMR expansion was significantly improved
using pre-conditioning with a minimal number of additional input-output samples.
B. Molecular and Solid State Materials Applications.
(1) Substituent Orderinga nd Interpolationi n Molecular Library Optimization[4]. The
substituent ordering problem in molecular libraries refers to identifying a rational
ordering for molecular moieties, such that coarse sampling and interpolation over
the full space of possible library molecules may be efficiently performed. This
work proposed a practical solution to the ordering problem, based on (a) coarse
sampling of the molecular substituents, (b) radial basis function interpolation over
the full space, and (c) the use of genetic algorithms to find rational moiety
orderings. The algorithm was also used to reorder and predict the glass transition
temperature Tg from laboratory data in a combinatorial polymer synthesis library.
(2) Material PropertiesO btained by Using the CorrelatedF unction Expansion for the
QuaternaryA lloy GaxInl_xPyAsl_y[5]. The band gap and the bond length of the
quaternary alloy GaxInl-xPyAsl-y were estimated over the entire composition
space using a HDMR interpolation technique. The HDMR band gap results
reproduced well the experimental data for 35 samples of the alloy with an average
error of 1.5%. The HDMR band gap variation of GaxInlixPyAsiy lattice matched
to InP was in good agreement with existing experimental data. This demonstrated
that HDMR can aid in the discovery of new materials with desired properties.
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C. Applications to Mathematical Modeling.
(1) Radiation Transport Simulation by Means of a Fully Equivalent Operational
Model[6J. The radiation transport component of atmospheric modeling codes is
typically a major contributor to the overall execution time. This research
introduced a HDMR as a high-speed replacement for traditional transport modules.
The input information to the HDMR was the atmospheric trace gases and
temperature profiles as a function of altitude, as well as the surface temperature and
albedo. The HDMR was shown to be better than 97% accurate over a broad input
variable range, while simultaneously being approximately 103 times faster than the
traditional radiation transport module it replaced.
(2) Optimal Control of Catalytic Methanol Conversion to Formaldehyde[7]. An
optimal control methodology was applied to find the heat and oxygen flux profiles,
distributed along the length of a plug flow reactor, for the conversion of methanol to
formaldehyde. A fully non-linear robustness analysis of the formaldehyde
production with respect to the catalyst model variables was performed by the use of
a HDMR. This representation is similar to the ANOVA decomposition used in
statistics but did not require an increase in the number of data points as the
dimensionality of the variable space increased.
(3) Global, Nonlinear Algorithm for Inverting Quantum-Mechanical Observations[8].
This work presented a global, nonlinear HDMR algorithm for extracting molecular
potentials from measurements of quantum-mechanical observables. The method
utilized a mapping technique to learn the relationship between a broad domain of
potentials and their resulting observables to facilitate the inversion. The inversion
of laboratory differential cross-section data for He + Ne scattering was performed.
The example provided a benchmark for the capabilities of the proposed algorithm to
stably reveal the full distribution of potentials consistent with the data.
(4) Constructing Global FunctionalM aps Between Molecular Potentials and Quantum
Observables[9]. The relationships that connect potential energy surfaces to
quantum observables can be complex and nonlinear. In this work, a HDMR
approach toward globally representing and exploring potential-observable
relationships using a functional mapping procedure was developed. It was
demonstrated that an observable's behavior could be learned reliably as a function
of the potential and any other variables needed to specify the quantum system.
(5) Achieving the Laboratory Control of Quantum Dynamics Phenomena Using
Nonlinear FunctionalM aps[]O]. This research introduced a HDMR algorithm for
achieving closed-loop laboratory control of quantum dynamics phenomena. The
procedure made use of nonlinear functional maps to exploit laboratory control data
for revealing the relationship between control fields and their effect on the
observables of interest. To demonstrate the operation of the proposed map-based
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control algorithm, two illustrations involving simulated population transfer
experiments were performed.
(6) Computationally Efficient Atmospheric Chemical Kinetic Modeling by Means of
High Dimensional Model Representation (HDMR)[I1]. For relieving the
computational burden of chemical kinetic calculations in air quality models, this
work presented an application of the HDMR method to greatly accelerate these
calculations. A HDMR expansion is used to calculate output species concentration
at a given reaction time based on the initial input species concentrations. The
application of the HDMR method focused on a photochemical box model study of
complex alkane/NOx/O photochemistry. The HDMR expansion was about 400
3
times faster than the original box-model.
Personnel Supported
Herschel Rabitz, Principal Investigator
Genyuan Li, Research Chemist
Omer Alis, Graduate Student (Ph.D. received, January, 2001)
John Geremia, Graduate Student (Ph.D. received, June, 2001)
New Discoveries, Inventions, or Patent Disclosures
"High Efficiency Mapping of Molecular Variations to Functional Properties"; Patent
Application filed January, 2001
"A Machine for Optimal Closed Loop Identification of Molecular Characteristics"; Patent
Application, Filed February, 2002.
Publications/References
[1] High Dimensional Model Representations, G. Li, C. Rosenthal, and H. Rabitz, J
Phys. Chem. A, 105, 7765-7777 (2001).
[2] Efficient Implementation of High Dimensional Model Representations, 0. Alis and
H. Rabitz, J Math. Chem., 29, 127-142 (2001).
[3] High dimensional model representations generated from low dimensional data
samples. I. mp-Cut-HDMR, G. Li, S.-W. Wang, C. Rosenthal, and H. Rabitz, J.
Math. Chem., 30, 1-30 (2001).
[4] Substituent Ordering and Interpolation in Molecular Library Optimization, N.
Shenvi, J.M. Geremia, and H. Rabitz, J. Phys. Chem., 2003, in press.
[5] Material Properties Obtained by Using the Correlated Function Expansion for the
Quaternary Alloy Gaxlnl-xPyAsl-y, K. Shim and H. Rabitz, J Korean Physical
Society, 2000, 37, 124-128 (2000).
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[6] Radiation transport simulation by means of a fully equivalent operational model, J.
Shorter, P. Ip, and H. Rabitz, Geophys. Res. Lett., 27, 3485-3488 (2000).
[7] Optimal Control of Catalytic Methanol Conversion to Formaldehyde, A. Faliks,
R.A. Yetter, C.A. Floudas, S. Bernasek,, M. Fransson, and H. Rabitz, J. Phys.
Chem. A, 105, 2099-2105 (2001).
[8] Global, nonlinear algorithm for inverting quantum-mechanical observations, J.M.
Geremia and H. Rabitz, Phys. Rev. A, 64, 022710-1-13 (2001).
[9] Constructing global functional maps between molecular potentials and quantum
observables, J.M. Geremia, H. Rabitz, and C. Rosenthal, J. Chem. Phys., 114, 9325-
9336 (2001).
[10] Achieving the Laboratory Control of Quantum Dynamics Phenomena Using
Nonlinear Functional Maps, J.M. Geremia, E. Weiss, and H. Rabitz, Chem. Phys.,
267,209-222 (2001).
[11] Computationally Efficient Atmospheric Chemical Kinetic Modeling by Means of
High Dimensional Model Representation (HDMR), S.W. Wang, P.G.
Georgopoulos, G. Li, and H. Rabitz, J. Geophys. Res., 2003, in press.
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